一类具功能性反应的Prey-Predator系统的周期解与稳定性

Existence and Stability Periodic Solution for Prey-Predator System with Functional Response

研究一类具HollingⅡ功能性函数的含扩散与时滞Prey-Predator系统,利用上下解及比较原理,通过周期抛物系统ui(t,x)/t-Aiui(t,x)=ui(t,x)[ai(t,x)-bi(t,x)ui(tx,x)](i=1,2)的周期解得到系统的上下解,证明了系统在对应的特征方程的主特征值σ1(a1)≥0,σ1(a2)>0时存在全局渐近稳定的平凡解(0,0),当σ1(a1)≥0,σ1(a2)<0时系统存在全局渐近稳定的半平凡解(0,Θ2(t,x)),当σ1(a1)<0,σ1(a2+1)≥0时系统存在全局渐近稳定的半平凡解(θ1(t,x),0),并获得当σ1(a1)<0,σ1(a2)<0时系统存在一对T-周期拟解的充分条件,且对任意的非负初值函数这对周期拟解构成此系统的一个吸引子。

The existence and stability of periodic solution in Prey-Predator system with diffusion,time-delay and Holling type II are investigated by using the method of upper and lower solutions and comparison principle.It is shown that the globally asymptotically stale trivial solution（0,0） when σ1（a1）≥0,σ1（a2）0,the globally asymptotically stale semi-trivial periodic solutions（0,Θ2（t,x））,（θ1（t,x）,0） when σ1（a1）≥0,σ1（a2）0 and σ1（a1）0,σ1（a2＋1）≥0 of the system by construction of a pair of upper and lower solution of parabolic periodic systemui（t,x）/t-Aiui（t,x）=ui（t,x）[ai（t,x）-bi（t,x）ui（tx,x）]（i=1,2）.It was obtained that the system have a pair of T-periodic quasi-solutions and the sector between the quasi-solutions is an attractor of the system with respect to every nonnegative initial function.